cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A293119 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(-x^i).

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 0, -2, -1, 1, 0, 0, -6, 1, 1, 0, 0, -6, -12, 19, 1, 0, 0, 0, -24, 0, 151, 1, 0, 0, 0, -24, -120, 240, 1091, 1, 0, 0, 0, 0, -120, -360, 2520, 7841, 1, 0, 0, 0, 0, -120, -720, 0, 21840, 56519, 1, 0, 0, 0, 0, 0, -720, -5040, 20160, 181440, 396271
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Examples

			Square array begins:
    1,   1,    1,    1, ...
   -1,   0,    0,    0, ...
   -1,  -2,    0,    0, ...
   -1,  -6,   -6,    0, ...
    1, -12,  -24,  -24, ...
   19,   0, -120, -120, ...

Links

Crossrefs

Columns k=0..2 give A293116, A293117, A293118.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
A(n,n-1) gives (-1)*A000142(n).
Cf. A293053.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, -add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 30 2017
  • Mathematica
    A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = -Sum[A[n-j, k] Binomial[n-1, j-1] j!, {j, k+1, n}];
A[, ] = 0;
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)

Formula

E.g.f. of column k: exp(x^(k+1)/(x-1)).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.

A293139 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} (-1)^j*x^(j*i)/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, -2, 0, 1, -1, -1, 0, 0, 1, -1, -1, 0, 0, 0, 1, -1, -1, -1, 0, 120, 0, 1, -1, -1, -1, 0, 0, 0, 0, 1, -1, -1, -1, 1, 20, 180, 5040, 0, 1, -1, -1, -1, 1, 20, 180, 0, 0, 0, 1, -1, -1, -1, 1, 19, 150, 1260, 10080, 0, 0, 1, -1, -1, -1, 1, 19
Offset: 0

Views

Author

Seiichi Manyama, Oct 01 2017

Keywords

Examples

			Square array begins:
   1,   1,  1,  1,  1, ...
   0,  -1, -1, -1, -1, ...
   0,  -2, -1, -1, -1, ...
   0,   0,  0, -1, -1, ...
   0,   0,  0,  0,  1, ...
   0, 120,  0, 20, 20, ...

Links

Crossrefs

Columns k=0..2 give A000007, A293140, A293141.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293135.

A334561 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 5, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 25, -41, 1, 1, -1, -1, -1, 1, 19, 31, -1, 1, -1, -1, -1, 1, 139, -209, 461, 1, 1, -1, -1, -1, 1, 19, 151, -2269, -895, -1, 1, -1, -1, -1, 1, 19, 871, -1429, 2801, -6481, 1, 1, -1, -1, -1, 1, 19, 151, 1091, -19039, 68615, 22591, -1
Offset: 0

Views

Author

Seiichi Manyama, May 06 2020

Keywords

Examples

			Square array begins:
   1,   1,    1,   1,   1,   1,   1, ...
  -1,  -1,   -1,  -1,  -1,  -1,  -1, ...
   1,  -1,   -1,  -1,  -1,  -1,  -1, ...
  -1,   5,   -1,  -1,  -1,  -1,  -1, ...
   1,   1,   25,   1,   1,   1,   1, ...
  -1, -41,   19, 139,  19,  19,  19, ...
   1,  31, -209, 151, 871, 151, 151, ...

Links

Crossrefs

Columns k=1..5 give A033999, A000321, A334562, A334564, A334565.
Main diagonal gives A293116.
Cf. A293669, A334568.

Formula

A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.

A293530 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/Product_{j > 0, j mod k > 0} exp(x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, -1, -7, 0, 1, -1, -1, 5, 25, 0, 1, -1, -1, -1, -23, -181, 0, 1, -1, -1, -1, 25, -41, 1201, 0, 1, -1, -1, -1, 1, -101, 1111, -10291, 0, 1, -1, -1, -1, 1, 139, -209, -6259, 97777, 0, 1, -1, -1, -1, 1, 19, -569, 251, -16015
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2017

Keywords

Examples

			Square array begins:
   1,    1,   1,    1,   1, ...
   0,   -1,  -1,   -1,  -1, ...
   0,    1,  -1,   -1,  -1, ...
   0,   -7,   5,   -1,  -1, ...
   0,   25, -23,   25,   1, ...
   0, -181, -41, -101, 139, ...

Links

Crossrefs

Columns k=1..3 give A000007, A293532, A293533.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293525.

Formula

E.g.f. of column k: exp((Sum_{j=1..k-1} x^j)/(x^k - 1)).

A293299 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} 1/(Sum_{j=0..k} x^(j*i)/j!).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -1, -6, 0, 1, -1, -1, 0, 24, 0, 1, -1, -1, -1, -6, -120, 0, 1, -1, -1, -1, 2, 30, 720, 0, 1, -1, -1, -1, 1, 10, 270, -5040, 0, 1, -1, -1, -1, 1, 20, 170, 0, 80640, 0, 1, -1, -1, -1, 1, 19, 140, 1050, 2520, -725760, 0, 1, -1
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2017

Keywords

Examples

			Square array begins:
   1,    1,  1,  1,  1, ...
   0,   -1, -1, -1, -1, ...
   0,    0, -1, -1, -1, ...
   0,   -6,  0, -1, -1, ...
   0,   24, -6,  2,  1, ...
   0, -120, 30, 10, 20, ...

Links

Crossrefs

Columns k=0..1 give A000007, A293300.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293135, A293301.

A293565 Expansion of e.g.f. Product_{k>=0} exp(-x^(3*k+1)).

Original entry on oeis.org

1, -1, 1, -1, -23, 119, -359, -4201, 58801, -359857, -2121839, 74851919, -784376999, -1072414201, 192596540137, -3167640042841, 9241215940321, 860635956717599, -21290802661288799, 157132294452113183, 5980504635868174921, -220612220726623344361
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Links

Crossrefs

E.g.f.: Product_{m>=0} exp(-x^(k*m+1)): A293116 (k=1), A293532 (k=2), this sequence (k=3), A293566 (k=4).
Cf. A293493.

Programs

  • Mathematica
    CoefficientList[Series[E^(x/(x^3 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(x^3-1))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(3*m+1)))))

Formula

E.g.f.: exp(x/(x^3 - 1)).
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^(n-k) * binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024

A293566 E.g.f.: Product_{m>=0} exp(-x^(4*m+1)).

Original entry on oeis.org

1, -1, 1, -1, 1, -121, 721, -2521, 6721, -378001, 5473441, -39972241, 199679041, -7005552841, 176899522801, -2186722497961, 17454339826561, -459473703430561, 16503993702423361, -306140370496394401, 3555223271216311681, -80917223353652470681
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Links

Crossrefs

E.g.f.: Product_{m>=0} exp(-x^(k*m+1)): A293116 (k=1), A293532 (k=2), A293565 (k=3), this sequence (k=4).
Cf. A293507.

Programs

  • Maple
    seq(factorial(k) * coeftayl(product(exp(-x^(4*m + 1)),m = 0..k), x = 0, k),k = 0..50); # Muniru A Asiru, Oct 15 2017
  • Mathematica
    CoefficientList[Series[E^(x/(x^4 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(x^4-1))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(4*m+1)))))

Formula

E.g.f.: exp(x/(x^4 - 1)).
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k+1)! * a(n-4*k-1). - Ilya Gutkovskiy, Feb 24 2022

A331688 E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 3, 17, 137, 1389, 16819, 236557, 3792753, 68326073, 1366917731, 30074632521, 721798881913, 18766625660197, 525460685327187, 15763716503597189, 504436925448024929, 17150818356045629937, 617428780939911647683, 23462281235407345160833
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2020

Keywords

Links

Crossrefs

Cf. A000165, A000166, A000354, A009940, A293116, A331689.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n) = -(n - 1)*(5*n - 8)*a(n - 2) + (-3 + 4*n)*a(n - 1) + 2*(n - 1)*(n - 2)^2*a(n - 3),a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 28 2020
  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[-x/(1 - x)]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! Subfactorial[n - k], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000166(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * k! * 2^k * A293116(n-k).
a(n) ~ n! * exp(-1) * 2^n. - Vaclav Kotesovec, Jan 26 2020
a(n) = (4*n-3)*a(n-1)-(n-1)*(5*n-8)*a(n-2)+2*(n-1)*(n--2)^2*a(n-3). - Robert Israel, Jul 28 2020

A330201 Expansion of e.g.f. Product_{k>=1} exp(-x^k) / (1 - x^k).

Original entry on oeis.org

1, 0, 1, 2, 21, 44, 1045, 2694, 74473, 421784, 8776521, 52518410, 1843753021, 11476952772, 387068115421, 4277646186254, 125796357803985, 1343857519264304, 53205974734877713, 621203524858308114, 25357790175078682981, 388778926109137187420
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2019

Keywords

Crossrefs

Cf. A000166, A000203, A051953, A053529, A293116, A330200.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[Exp[-x^k]/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (n - 1)! Sum[(DivisorSigma[1, k] - k) a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of A000166.
E.g.f.: exp(Sum_{k>=1} (sigma(k) / k - 1) * x^k), where sigma = A000203.
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(cototient(k)/k), where cototient = A051953.
a(0) = 1; a(n) = (n - 1)! * Sum_{k=1..n} (sigma(k) - k) * a(n-k) / (n - k)!.
a(n) = Sum_{k=0..n} binomial(n,k) * A293116(k) * A053529(n-k).
a(n) ~ sqrt(-1/Pi + Pi/6) * n^(n - 1/2) / (2 * exp(n - 1/2 - sqrt(2*(-6 + Pi^2)*n/3))). - Vaclav Kotesovec, Aug 09 2021

A346770 Expansion of g.f. Product_{k>=1} (1 - x^k)^phi(k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, -1, -1, -1, 0, 0, 3, 1, 4, 2, 3, -5, 1, -13, -5, -13, -6, -22, 12, -12, 35, 17, 59, 11, 101, -1, 81, -35, 45, -165, 29, -311, -69, -383, -57, -501, 181, -501, 425, -191, 990, -70, 1844, 64, 2305, 183, 2625, -951, 2897, -2701, 1845, -4851, 664, -8824, 670, -12366, 269, -14137, 2884
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2021

Keywords

Links

Crossrefs

Convolution inverse of A061255.
Cf. A000010, A057660, A293116, A299069.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^eulerphi(k)))
  • PARI
    N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, sigma(k^2, 2)/sigma(k^2)*x^k/k)))

Formula

G.f.: exp(-Sum_{k>=1} A057660(k) * x^k/k).
a(0) = 1, a(n) = -(1/n) * Sum_{k=1..n} A057660(k) * a(n-k) for n > 0.
Showing 1-10 of 10 results.

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